On the spectrum of some cyclically and transitively oriented $\mathcal{H}$-designs

Source abstract

This paper investigates oriented $\mathcal{H}$-designs on complete uniform hypergraphs of rank 3, focusing in particular on the spectrum of existence and on the construction of some cyclically and transitively oriented $P^{(3)}(2,4)$-designs and $P^{(3)}(1,5)$-designs, namely $BCP^{(3)}(2,4)$-designs and $BCP^{(3)}(1,5)$-designs for cyclically oriented ones, $BTP^{(3)}(2,4)$-designs and $BTP^{(3)}(1,5)$-designs for transitively oriented ones. In the appendix, we provide the python code to obtain the explicit realization of the $BCP^{(3)}(2,4)$-designs on $v$ vertices. Moreover, the structure of this algorithm, with suitable modifications, can be generalized to the other three structures as well.Accepted: 5 November 2025